A New Method for Optimal Solution of Intuitionistic Fuzzy Transportation Problems via Generalized Trapezoidal Intuitionistic Fuzzy Numbers

In this paper, we introduce the new method for solving the intuitionistic fuzzy transportation problem (IFTP), by using north-west corner method and modified distribution method to find the optimal solution for IFTP.


Introduction
In 1956, Zadeh [1] firstly defined the concept of fuzzy set theory. The concept of an intuitionistic fuzzy set was proposed by Atanassov in 1986 [2]. This concept referred to the reflection of the relation among '1 minus the degree of membership', 'the degree of non-membership' and 'the degree of hesitation'. The intuitionistic fuzzy set was rasterised by the degree of membership and the degree of non-membership. The intuitionistic fuzzy set had more abundant and flexible than the fuzzy set with uncertain information. Many researchers have also used fuzzy and intuitionistic fuzzy set for solving real world optimisation problems such as transportation problem.
The transportation problem is a special kind of optimisation problem. Transportation problem is interested in finding the least total transportation cost of goods in order to satisfy demand at destinations using available supplies at the sources. In usual, transportation problems are solved with the hypothesis that values of supplies and demands and the transportation costs are specified in a precise way. In the real world, in many cases, the decision-maker has no crisp information about the coefficients belonging to the transportation problem. In this situation, the corresponding elements defining the problem can be formulated by mean of fuzzy set, and the fuzzy transportation problem appears in a natural way. In 1941, Hitchcock [3] originally developed the basic transportation problem. Dantzig [4] applied linear programming to solving the transportation problem. Several authors have carried out an examination about fuzzy transportation problem [5][6][7][8][9]. Moreover, several authors have used intuitionistic fuzzy set theory for solving transportation problems. Hussain and Kumar [10] investigate the transportation problem with the aid of triangular intuitionistic fuzzy numbers (TIFN). Pramila and Uthra [11] presented optimal solution of an IFTP. Antony et al. [12] studied method for solving the transportation problem by using TIFN. Singh and Yadav [13] discussed new approach for solving IFTP of type-2 where the supply, demand are fixed crisp numbers and the cost is TIFN.
In this paper, we using a linear ranking function for generalised trapezoidal intuitionistic fuzzy numbers (GTrIFNs) to find the IBFS and optimal solution of GTrIFNs based on the allocation of demands and availabilities are real numbers and cots are GTrIFNs. This paper is organised as follows. Section 2 gives the concept of mathematics preliminaries. Section 3 presents ranking of GTrIFN. Section 4 describes a mathematics formulation for IFTP. Section 5 details some numerical example. In the final section, the paper is concluded in Section 6.

Mathematical Preliminaries
In this section, we give some basic definitions and concepts of cut sets of trapezoidal intuitionistic fuzzy number (TrIFN).

Some Definitions of TrIFNs
Definition 2.1: [1]: Let X be an arbitrary nonempty set of the universe. A fuzzy set A in X is a function with domain X and values in [0, 1]. If A is a fuzzy set and x ∈ X, then the function value μ A (x) is called the membership function of x in A. A fuzzy set can be written as order pair, given by {x, Definition 2.2: [2]: Let X be an arbitrary nonempty set of the universe. If there are two mapping on the set X: The μ A and ν A are called determining and intuitionistic fuzzy set A on the universal set X, denote by {x; μ A (x), ν A (x)|x ∈X} we called μ A and ν A are membership function and nonmembership function of A, respectively. μ A (x) and ν A (x) are called the membership degree and nonmembership degree of an element x belonging to A ⊆ X, respectively. IF(X) is called the set of the intuitionistic fuzzy set on the universal set X. Figure 1 if its membership and nonmembership functions are defined as follows:

Definition 2.4: A TrIFN
π A (x) is called the hesitancy degree of an element x ∈ A. It is the degree of indeterminacy membership of the element x to A.
From Definition 2.4, it is obvious that μ A (x) + ν A (x) = 1 for any x ∈ R if t A = 1 and z A = 0. Hence, the TrIFN A = (l, c, d, r); t A , z A degenerates to A = (l, c, d, r); 1, 0, which is a trapezoidal fuzzy number (TrFN) [14]. Therefore, the concept of the TrIFN is generalisation of that of the TrFN.
From A = (l, c, d, r); t A , z A if c = d = p then A = (l, p, r); t A , z A that is A = (l, p, r); t A , z A is a TIFN, which is particular case of TrIFN. Likewise to algebraic operations of TIFN and TrIFN are defined as follows.
Definition 2.5: Let A = (l 1 , c 1 , d 1 , r 1 ); t A , z A and B = (l 2 , c 2 , d 2 , r 2 ); t B , z B be two GTrIFNs with t A = t B , z A = z B and γ = 0 be any real number. Then, the algebraic operations of GTrIFNs are defined as follows: where the symbols and is the minimum operator and ∨ is the maximum operator.

Cut Sets of TrIFN
Definition 2.6: [15]: A (α, λ)− cut set of A = (l, c, d, r); t A , z A is a crisp subset of R, which is defined as follows: Definition 2.7: [15]: The α− cut set and λ-cut set of A = (l, c, d, r); t A , z A are a crisp subset of R, which is defined as follows: Using the membership function of A = (l, c, d, r); t A , z A and Definition 2.7 such that A α = {x|μ A (x) ≥ α} and A * λ = {x|ν A (x) ≤ λ} are closed interval and calculated as follows: and respectively.

Ranking of TrIFN
This section briefly reviews the ambiguities and the accuracy function of a GTrIFN. and where h(α) and g(λ) satisfy the following conditions: Let A be an arbitrary IFN. The ambiguities for IFN A for membership and nonmembership functions are denote by V (µ A ) and V (ν A ), respectively. respectively. V(μ A ) and V(ν A ) are defined by and Next, we find score, accuracy and ambiguities function of a GTrIFN. Let a GTrIFN A = (l, c, d, r); t A , z A the score function of a GTrIFN A for membership and non-membership functions can be written as follows: from Equations (10), (12) and h(α) = α, we get Similarly, from Equations (11), (13) and g(λ) = λ, we have The accuracy function of a GTrIFN A is denoted by from Equations (10), (14) and h(α) = α, we get Similarly, from Equations (11), (15) and g(λ) = λ, we get The accuracy function of a GTrIFN A is denoted by In the same way, if γ < 0, β < 0 we can prove (γ A + B) = γ (A) + (B).
Therefore, is a linear function. LetA = (a 1 , a 2 , a 3 , a 4 ); t A , z A and B = (b 1 , b 2 , b 3 , b 4 ); t B , z B be GTrIFNs with t A = t B and z A = z B . The ambiguities function : GIF(R) → R is a linear function.

Theorem 3.2:
(The rest of the proof is similar to proof of Theorem 3.1). A = (a 1 , a 2 , a 3 , a 4 ); t A , z A and B = (b 1 , b 2 , b 3 , b 4 ); t B , z B be GTrIFNs. The ranking order of AandB is stipulated as follows:

Mathematical Formulation for IFTP
This section, first introduces the mathematical formulation of the IFTP. Later, we find IBFS by NWCM and we use MODIM for finding optimal solution. The mathematical formulation of the IFTP is of the following form: x ij ≥ 0 for all i and j where c ij be GTrIFN cost of transportation one unit of the goods from i th source to the j th destination. x ij be the quantity transportation from i th source to the j th destination, is shown Table 1.
Here, a i be the total availability of the goods at i th source. b i be the total demand of the goods at j th destination.
n j=1 x ij c ij be total intuitionistic fuzzy transportation cost. If m i=1 a i = n j=1 b j then IFTP is said to be balanced. If m i=1 a i = n j=1 b j then IFTP is said to be unbalanced (Table 1) where A be an m × n matrix, X be an n − vector, b be an m − vector, and c = (c 11 , c 12 , . . . , c 1n , . . . , c m1 , . . . , c mn ) T .

Algorithm to Find an Initial Basic Feasible Solution (IBFS) of IFTP
In this section, we use intuitionistic fuzzy NWCM to compute IBFS of IFTP.
Step 1: Set up the formulated intuitionistic fuzzy linear programming problem into the tabular form know as intuitionistic fuzzy transportation table (IFTT). An we approximate cost by GTrIFNs.
Step 2: Examine that the IFTP is balanced or unbalanced, if unbalanced, make it balanced.
Step 3: Choose the north-west corner cell (NWCC) of the IFTT. Let it be the cell(i, j).Find min(a i , b j ), then allocate x ij = a i in the (i, j)th cell of m × n IFTT. Delete the ith row to obtain a new IFTT of order (m − 1) × n. Replace b j by b j − a i in obtained IFTT. Go to step 4. (a i , b j ), then allocate x ij = b j in the (i, j)th cell of m × n IFTT. Delete the jth column to obtain a new allocate IFTT of order (m) × (n − 1).Replace a i by a i − b j in obtained IFTT. Go to step 4. case (iii) If a i = b j , then either follow case(i) or case(ii) but not both together. Go to step 4.
Step 4: Calculate the penalties for the reduced IFTT obtain in step 3. Repeat step 3 until the IFTT is reduced to 1 × 1.
Step 5: Allocate all x ij in the (i, j)th cell of the given IFTT.
Step 6: The obtained IBFS and initial intuitionistic fuzzy transportation cost are x ij and m i=1 n j=1 x ij c ij respectively.

Modified Distribution Method for Finding Optimal Solution
In this section, we use generalised intuitionistic modified distribution method (GIMODIM) to find the optimal solution for IFTP. Algorithm of GIMODIM is illustrated as follows: Step 1: Find IBFS by propose IFNWCM.
Step 2: Compute IF dual variables u i and v j for all row and column, respectively, satisfying (c ij ) = (u i ⊕ v j ) for all occupied cell. To start with. take any v j or u i as (−1, 0, 0, 1; 1, 0). Step 3: For unoccupied cells, find opportunity e ij = e ij ij , where ij = u i ⊕ v j .
Step 4: Consider valued of (e ij ). case (i) IBFS is the intuitionistic fuzzy optimal solution, if (e ij ) ≥ 0 for all unoccupied cells. case (ii) IBFS is not the intuitionistic fuzzy optimal solution, for at least one (e ij ) < 0. Go to step 5.
Step 5: Choose the unoccupied cell for the most negative value of (e ij ).
Step 6: We construct the closed loop below. At first, start the closed loop with choose the unoccupied cell and move vertically and horizontally with corner cells occupied and come back to choose the unoccupied cell to complete the loop. Use sign '+' and '−' at the corners of the closed loop, by assigning the '+' sign to the selected unoccupied cell first.
Step 7: Look for the least allocation value from the cells which have '−'sign. After that, allocate this value to the choose empty cell and subtract it to the other occupied cell having '−' sign and add it to the other occupied cells having '+' sign.
Step 8: Allocation in Step 7 will result an improved basic feasible solution (BFS).
Step 9: Test the optimality condition for improved BFS. The process is complete when (e ij ) ≥ 0 for all the unoccupied cell.
For transportation costs from island to plant are as follows: (unit: 10 Baht per one kilogram of bird's nest).
From table 4, we will find out the minimum cost of total fuzzy transportation. Since 3 i=1 a i = 3 j=1 b j = 125, the FTP is balanced. Finding IBFS of IFTP by IFNWCM. Now, transfer this allocation to the FTT. The first allocation is shown in Table 3 and the final allocation is shown in Table 4.
Therefore Now, we apply GIMODIM to compute the optimal solution. Algorithm of modified distribution method as shown in Section 4.2.
Firstly, we compute intuitionistic fuzzy dual variables u i and v j for each row and column, respectively, satisfying u i ⊕ v j = c ij for each occupied cell. Therefore, let v 1 = (−1, 0, 0, 1); 1, 0 .  From above, we found that the value of e 13 is most negative, so IBFS is not intuitionistic fuzzy optimal.
Check e ij again, if e ij ≥ 0 for all unoccupied cells, then the solution is intuitionistic fuzzy optimal solution. If e ij < 0, go to Step 5.
Next, improved Basic Feasible Solution.
Let v 1 = (−1, 0, 0, 1); 1, 0 . For each occupied cell, u i ⊕ v j = c ij , we compute    Hence, we observe that From above, we found that the value of e ij ≥ 0 for all unoccupied cells, so optimal solution is x 11 = 5, x 12 = 5, x 13 = 25, x 21 = 40, x 32 = 50 shown in Table 6, and the minimum transportation intuitionistic fuzzy cost is The minimum transportation intuitionistic fuzzy cost can be interpreted as follows: the minimum transportation intutionistic fuzzy costs stay in the ranges [380, 1660], when (α, λ) = (0.5, 0.3). That is, the degree of acceptance of the transportation cost for the decision making increases if the cost increases from 380 to 770. The degree of acceptance of the transportation cost for the decision making is stationary when the costs are in the range 770-1120, while it decreases if the cost in increases from1120 to 1660. The transportation cost is totally acceptable if transportation cost stays in the ranges [770,1120]. The degree of non-acceptance of the transportation cost for the decision making decreases if the cost increases from 380 to 770. The degree of un-acceptance of the transportation cost for the decision making is stationary when the costs are in the range 770-1120, while it increases if the cost increases from 1120 to 1660.

Conclusion
In this paper, we are defined a new concept of linear ranking function for GTrIFNs. This new method is proposed to find the IBFS and the optimal solution of GTrIFNs based on the both demands and availabilities are real numbers. In addition, the cost is always GTrIFNs under the condition of the linear transportation problem. The advantages of this method can be used to solve for all kinds of IFTP, whether triangular fuzzy number, TrFN, TIFN, TrIFN or GTrIFN which this method is obtained solution is always optimal. Moreover, this method can use both the maximum and minimum values of an objective function. However, this method has a limit for the linear multi-objective transportation problem and including other (nonlinear) shapes for membership functions, such as exponential membership function and hyperbolic membership function etc.